This was such a clean solution, had to share it. After a few days, the adhesive on my LED strips failed, probably because the painted 2x4 lumber was too rough of a surface for them. I found a good solution was to use these little "staples" used in the US to support NM style mains electrical cable.
They are sometimes called Romex staples or NM staples, but they are actually just a plastic holder with two nails. They make them in multiple sizes, the 1/2" size worked fine for me. They look nice and hold the LED strips securely, I just installed one about every 8" or so making sure not to cover an LED. Below you can see the result: